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Matrix concentration inequalities for dependent binary random variables

Radosław Adamczak, Ioannis Kavvadias

TL;DR

This work extends matrix concentration to dependent binary variables under $\ell_∞$-independence, SCP, and SRP. It develops Bernstein-type bounds for $Z = \sum_{i=1}^n X_i A_i$ with $A_i$ symmetric, and extends to general matrix-valued Lipschitz functions under SCP, plus a decoupling framework for Banach-space quadratic forms that yields submatrix-norm bounds under SRP sampling. The results generalize and strengthen prior independent-matrix inequalities and prior dependent-structure bounds, enabling rigorous analysis in combinatorial, statistical physics, and sampling contexts where inclusion variables interact. The work integrates two-stage sampling, coupling, and operator-norm techniques to produce practical tail estimates and sampling bounds with broad applicability in dependent binary models.

Abstract

We prove Bernstein-type matrix concentration inequalities for linear combinations with matrix coefficients of binary random variables satisfying certain $\ell_\infty$-independence assumptions, complementing recent results by Kaufman, Kyng and Solda. For random variables with the Stochastic Covering Property or Strong Rayleigh Property we prove estimates for general functions satisfying certain direction aware matrix bounded difference inequalities, generalizing and strengthening earlier estimates by the first-named author and Polaczyk. We also demonstrate a general decoupling inequality for a class of Banach-space valued quadratic forms in negatively associated random variables and combine it with the matrix Bernstein inequality to generalize results by Tropp, Chrétien and Darses, and Ruetz and Schnass, concerning the operator norm of a random submatrix of a deterministic matrix, drawn by uniform sampling without replacements or rejective sampling, to submatrices given by general Strong Rayleigh sampling schemes.

Matrix concentration inequalities for dependent binary random variables

TL;DR

This work extends matrix concentration to dependent binary variables under -independence, SCP, and SRP. It develops Bernstein-type bounds for with symmetric, and extends to general matrix-valued Lipschitz functions under SCP, plus a decoupling framework for Banach-space quadratic forms that yields submatrix-norm bounds under SRP sampling. The results generalize and strengthen prior independent-matrix inequalities and prior dependent-structure bounds, enabling rigorous analysis in combinatorial, statistical physics, and sampling contexts where inclusion variables interact. The work integrates two-stage sampling, coupling, and operator-norm techniques to produce practical tail estimates and sampling bounds with broad applicability in dependent binary models.

Abstract

We prove Bernstein-type matrix concentration inequalities for linear combinations with matrix coefficients of binary random variables satisfying certain -independence assumptions, complementing recent results by Kaufman, Kyng and Solda. For random variables with the Stochastic Covering Property or Strong Rayleigh Property we prove estimates for general functions satisfying certain direction aware matrix bounded difference inequalities, generalizing and strengthening earlier estimates by the first-named author and Polaczyk. We also demonstrate a general decoupling inequality for a class of Banach-space valued quadratic forms in negatively associated random variables and combine it with the matrix Bernstein inequality to generalize results by Tropp, Chrétien and Darses, and Ruetz and Schnass, concerning the operator norm of a random submatrix of a deterministic matrix, drawn by uniform sampling without replacements or rejective sampling, to submatrices given by general Strong Rayleigh sampling schemes.

Paper Structure

This paper contains 13 sections, 19 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Basic notation
  4. Assumptions on random vectors
  5. Main results
  6. Matrix Bernstein inequality for $\ell_\infty$-independent random vectors
  7. Concentration for Lipschitz functions of random vectors withthe SCP
  8. A decoupling inequality with application to random submatrices
  9. Proofs
  10. Preliminary lemmas
  11. Proof of results from Section \ref{['sec:Bernstein']}
  12. Proofs of results from Section \ref{['sec:SCP']}
  13. Proofs of results from Section \ref{['sec:decoupling']}

Key Result

Theorem 3.1

Let $X = (X_1,\ldots,X_n)$ be an $\{0,1\}^n$-valued random vector and consider $A_1,\ldots,A_n \in \mathcal{M}_{sym}^{d\times d}$. Define the random variable Assume that $X$ is $k$-homogeneous and one-sided $\ell_\infty$-independent with parameter $D$. Then, for some absolute constant $C$ and all $t > 0$, where $\sigma^2 = \|\mathbb{E} \sum_{i=1}^n X_i A_i^2\|$ and $R = \max_{i\le n} \|A_i\|$.

Theorems & Definitions (43)

  • Definition 2.1: $\ell_\infty$-independence
  • Remark 2.2
  • Definition 2.3: Negative association
  • Definition 2.4: Rayleigh Properties
  • Definition 2.5: Stochastic covering property
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Corollary 3.5
  • ...and 33 more